How do i snap a copied part of a mesh to the parent of the mesh, wiht a distance created by edge?
What you need to do is enable Snapping.

After you have duplicated your mesh:

1: At the bottom of the 3D window enable the Snapping tool. Then click on the icon to the right of that and choose the the element to snapt to, in this case choose Vertex. To the right of that is some snap target options, choose Active.

2: With everything deselected, right mouse select the vertex on the duplicated mesh that you want to be able to snap to any other vertex. 
This vertex will be the Active vertex. (it is highlighted in red in the gif below). Now select the rest of the duplicated mesh by hovering your mouse over part of it and hit the L key.

3: G key to move the duplicated mesh about and you will find that the Active vertex of the duplicate will snap to the other vertices.

Don't forget to disable the snapping tool whnen you have finished with it .

Something you may not be aware of is that a mesh face only has one side. You can see this if you enable Backface culling from the Properties panel.

You should now be able to see that the duplicated mesh has its faces facing the wrong way. Facing inwards! this will create shading problems later on so you need to Flip the faces so that they will be on the outside when you join the edges of both meshes.

To do this: With only the duplicate mesh selected, open the Tools panel on the right and then open the Shading tab. From the Normals menu choose the Flip Direction option.

To join the edges of the original and duplicated mesh: In Edge select mode select the outer edges of both meshes then hit the W key to open the Specials menu. From there choose the Bridge Edge Loops option. **